Revision: Fuzzy logic
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1 Fuzzy Logic 1
2 Revision: Fuzzy logic Fuzzy logic can be conceptualized as a generalization of classical logic. Modern fuzzy logic aims to model those problems in which imprecise data must be used or in which the rules of inference are formulated in a very general way making use of diffuse categories. Fuzzy logic is a way of processing data by allowing partial set membership. 2
3 Revision: The fuzzy set concept The difference between classical and fuzzy sets is established by introducing a membership function. X = {x1,x2,...,xn}, if x1 belongs to the set Classical: Z()=(1,0,0,...,0) Fuzzy: Z()=(0.7,0,0,...,0) Definition: Let X be a classical universal set. real function μ : X [0, 1] is called the membership function of and defines the fuzzy set of X. This is the set of all pairs (x, μ(x)) with x X. 3
4 Revision: example Definition: Let V be the variable (quality of service, amount of the tip, etc.), X the range of values of the variable(for example, between 0 and 30 for the amount of tip) and Tv a finite or infinite set of fuzzy sub-sets. linguistic variable corresponds to the triplet (V, X, Tv). 4
5 Revision: The fuzzy set concept Example: age The membership of a person to the set of mature goes slowly from 0 to 1 Membership functions for the concepts young, mature and old 5
6 Revision: The membership function The definition of the membership functions is a very delicate point in the design of the Fuzzy Control, because the only restriction that a membership function has to satisfy is that its values must be in the [0,1] range. 6
7 Revision: The membership function fuzzy set can therefore, unlike a crisp one, be represented by an infinite number of membership functions. Membership functions can have diverse forms according to their definition: Triangular, Trapezoidal, Gaussian, Sigmoid... 7
8 Revision: The membership function s a general rule, we can sort out the methods of determination of a membership function into the following three categories : utomatic methods No expert is available. The most common methods are based on neural networks and/or on genetic algorithms. Statistical methods In the statistical methods some data expressed in the form of frequency histograms or other probability curves are used as a base to construct a membership function. Psychological methods This kind of strategy is often referred to as natural extraction of membership functions. For example: : the expert just has to choose the central value and the curve slope on either side. 8
9 9 Revision: Notation x U x x U x Ker x U x Supp U x x U x x x x x U = cut() - 1 ) ( 0 ) ( sup = h(), ) ( 0,1 : h= height Supp= support Ker=Kernel (Noyau in french)
10 Revision Degree of membership h()=1 Ker()=15 Supp()=]10,20[ 0.5-cut()=[12.5,17.5] Output variable 10
11 Revision fuzzy set is completely determined by its membership function. It also covers the case in which X is not a finite set. fuzzy set with the finite set of support {a1,a2,...,am} can be described in the following way = μ1/a1 + μ2/a2 + + μm/am Example: X = {x1,x2,x3}. The classical subsets ={x1,x2} and B = {x2,x3} can be represented as: =1/x1 +1/x2 +0/x3 B =0/x1 +1/x2 +1/x3. 11
12 Revision 12
13 13 Revision: logic operators X x * µb(x) µ(x) (x) µ Sugeno X x µb(x)), max(µ(x) = (x) µ Mamdani X x µb(x) µ(x) = (x) µ X x (x) * µ (x) µ - (x) µ (x) µ = (x) µ : Sugeno X x (x)) µ, (x) min(µ = (x) µ : Mamdani X x (x) µ (x) µ = (x) µ B B B B B B B B B B
14 14 Revision: logic operators B = B) ( to s correspond B = B) ( B = B) ( to s correspond B = B) ( X x (x) µ - 1 = (x) µ (x) µ = (x) µ c c c c c c c c
15 Revision: logic operators c X Example : c = max(0.4,1-0.4) 1 c Example : c = min(0.4,1-0.4) 0 15
16 Revision: Fuzzy inference Knowledge that can only be formulated in a fuzzy, imprecise manner can be captured in rules that can be processed by a computer. Example: R1: If (temperature = cold) then heat. R2: If (temperature = normal) then maintain. R3: If (temperature=warm) then reduce power. Temperature = cold/0.5+normal/0.3+warm/0.0. ction= heat/0.5+maintain/0.3+reduce/
17 Revision: Fuzzy inference Temperature = cold/0.5+ normal/0.3+warm/0.0. ction= heat/0.5+maintain/0.3+ reduce/
18 Revision: Fuzzy logic Knowledge Base 1-Set of rules 2-Fuzzy operators Input variables Fuzzy Sets 3-Inference method Defuzzificator Output variables e 1 e 2 e 3 Fuzzification Inference motor Defuzzification u 1 u 2 u 3 18
19 Fuzzy numbers and inverse operation a fuzzy controller operates, in general, in three steps: a) measurement is transformed into a fuzzy category using the membership functions of all defined categories; b) ll pertinent inference rules of the control system are evaluated and a fuzzy inference is produced; c) In the last step the result of the fuzzy inference is transformed into a crisp value. 19
20 Defuzzification Defuzzifier transforms the fuzzy inferences into a specific control variable. Op1 Op2 Op3 Op4 20
21 Defuzzification Defuzzifier transforms the fuzzy inferences into a specific control variable. i.e. it calculates each associated output and put them into a table: the lookup table. s with all fuzzy operators, the fuzzy system designer must choose among several possible defuzzification. We briefly present the two main methods of defuzzification : - Mean of Maximum (MoM) - Centre of Gravity (COG). 21
22 Exemple This example will be the decision of the amount of the tip at the end of a restaurant meal, depending on the quality of service and the quality of the food. We need to redefine membership functions for each fuzzy sub-set of each of our three variables: Input 1 : quality of service. Sub-sets : bad, good and excellent. Input 2 : quality of the meal. Sub-sets : execrable and delicious. Output : amount of the tip. Sub-sets : low, average and high. 22
23 Example Linguistic variable «quality of service» V = quality of service X = [0, 10] Tv = {bad, good, excellent} 23
24 Exemple Linguistic Variable «quality of food» V = quality food X = [0, 10] Tv = {execrable, delicious} 24
25 Exemple Linguistic variable «amount of the tip» V = amount of the tip X = [0, 30] Tv = {low, average, high} 25
26 Exemple Let us see what give the SUGENO operators if we have the quality of service = 1.1 and the quality of food = With the membership functions we defined above, we have: bad execrable (1,1) (1,19 0,7642 For the SUGENO operator OR, μ B (x) = μ (x) + μ B (x) μ (x).μ B (x), we have: ) 0, bad execrable bad ( 1,1) (1,19 ) (1,1) execrable bad execrable (1,19 ) 0,985 For the SUGENO operator ND, μ B (x) = μ (x).μ B (x) wa have: bad execrable bad ( 1,1) (1,19 ) execrable 0,
27 Example We determine the output variable: bad execrable bad ( 1,1) (1,19 ) (1,1) execrable bad execrable (1,19 ) 0,985 bad execrable bad ( 1,1) (1,19 ) execrable 0,
28 Exemple Let us now see what we would have obtained if we had used Zadeh MIN/MX operators: bad execrable (1,1) (1,19 0,7642 ) 0, By applying the MX rule for the OR operator, we obtain : bad execrable Max ( bad (1,1), execrable (1,19 )) 0, nd by applying the MIN rule for the ND operator, we obtain: bad execrable Min ( bad (1,1), execrable (1,19 )) 0,
29 Example Here is what these calculations give graphically : bad execrable Max ( bad (1,1), execrable (1,19 )) 0, bad execrable Min ( bad (1,1), execrable (1,19 )) 0,
30 Example In summary, we find that the choice of operators ND and OR affect our results. The following summary table highlights these differences: Opérateurs de Zadeh Min/Max ET Min ( ( x ), ( x )) B 0,7642 OU Max ( ( x ), ( x )) B 0, Opérateurs de Sugeno ET OU ( x ) ( x ) 0,7158 ( x ) ( x ) ( x ) ( x ) 0, 985 B B B This is the designer of fuzzy system will choose operators based on the knowledge he has on the system that must be modeled. 30
31 Decision Matrix s we defined the fuzzy operators ND, OR and NOT, the premise of a fuzzy rule may well be formed by a conjunction (ND) or disjunction (OR) of fuzzy propositions. The rules of a fuzzy system is called the decision matrix. Here's one example of our tips: If the service est bad or the food is execrable If the service is good If the service is excellent or the food is delicious then the amount of tip is low. then the amount of tip is average. then the amount of tip is high 31
32 Example We will now apply all three rules of our decision matrix. If the service est bad or the food is execrable If the service is good If the service is excellent or the food is delicious then the amount of tip is low. then the amount of tip is average. then the amount of tip is high For each rule, we obtain an intermediate result. pplication example of all three rules of our decision matrix with inputs(service = 7,83 ; food = 7,32) Rules for using the OR operator (Rules 1 and 3), we use the MX operation (corresponding operators Zadeh MIN/MX). 32
33 Example We aggregate these three intermediate results into a final result by simply taking the maximum. We see the final result at the bottom right in the diagram below. fuzzy rules is still a vague or fuzzy element. 33
34 Defuzzification: Mean of Maximum (MoM) Defuzzification MM defines the output (decision tip amount) as the average of the maximum of the fuzzy set resulting from the aggregation of findings. The average maximum abscissa is defined as: Decision s s ydy dy or S y m R, ( y ) SUP ( ( y )) m y R and R is the fuzzy set resulting from the aggregation of findings. 34
35 Defuzzification MoM Defuzzification with Maximum verage méthod(m) 35
36 Defuzzification COG The defuzzification COG is most commonly used. It sets the output to correspond to the abscissa of the center of gravity of the surface of the membership function from fuzzy set characterizing the aggregation finding. The abscissa of the center of gravity of the surface is calculated as follows: Decision s s y. ( y ) dy ( y ) dy With s = R, i.e. all possible output value 36
37 La défuzzification COG Defuzzification with the center of gravity method(cog) 37
38 Defuzzification COG vs MoM COG defuzzification is usualy used in fuzzy control. The M defuzzification is used to discriminate an output value. (e.g., pattern recognition) The definition of COG defuzzification avoids discontinuities that could appear in the defuzzification M, but is more complex and requires greater computation. 38
39 Defuzzification COG vs MoM Some works try to improve performance by seeking other equally effective methods but with lower computational complexity. s we see the two figures show the defuzzification methods M and COG applied to our example, the choice of this method has an important effect on the final decision. 39
40 Defuzzification: The Lookup Table Defuzzification needs to be performed for each subset of a membership function, both inputs and outputs. For instance, in the air conditioner system, one needs to perform defuzzification for each subset of temperature input such as LOW, MEDIUM and HIGH based on the associated fuzzy rules. The defuzzification result for each subset needs to be stored in the associated location in the lookup table according to the current temperature and temperature change rate. 40
41 Conclusion In the definitions, we have seen that the designer of the fuzzy system must make a number of important choices. These choices are based primarily on the advice of the expert or the statistical analysis of data, in particular to define the membership functions and the decision matrix. 41
42 Conclusion Here is a synopsis of a fuzzy system: Knowledge Base 1-Set of rules 2-Fuzzy operators Input Variables e 1 e 2 e 3 Fuzzy Sets 3-Inference method Inference Motor Defuzzification Defuzzificator Fuzzification Output Variables u 1 u 2 u 3 42
43 Conclusion In our example : the input is the 'quality of service = 7.83 and the 'quality of food' = 7.32; Fuzzificator corresponds to the 3 linguistic variables of 'service quality','food quality'and 'tip amount'; the inference motor comprises selecting fuzzy operators; the fuzzy knowledge base is the set of fuzzy rules; Defuzzificator is the part where the defuzzification method is applied; The output is the final decision: 'the tip amount' =
44 Conclusion It is interesting to see all the decisions based on each variable with our fuzzy inference with respect to the set of decisions that we would get using conventional logical type system: ll decisions of a fuzzy system 44
45 Conclusion ll decisions of a system based on classical logic, can generate only linear surfaces 45
46 Conclusion Thus, the power of fuzzy logic is to make possible the development of inference systems whose decisions are seamless, flexible and non-linear, closer to human behavior than is classical logic. In addition, the rules of the decision matrix are expressed in natural language. This has many advantages, such as including knowledge of an expert in the middle of a decision-making to model finely the aspects of natural language. 46
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